Hello there,
The JR Pass covers travel from Tokyo - Hakone - Osaka - Nagasaki and everything in between. You can make the stops whenever you feel like it.
Travel from Tokyo - Nagasaki in one go using the JR Pass is just below 500 minutes. So it's a rather long ride. Both Hakone (Odawara station) Osaka, Hiroshima and Fukuoka are directly on the route. So you could make a stop in between where you like.
The flight (air) time from Tokyo Haneda to Nagasaki is about 110 minutes, however keep in mind that you still have to go to the airport, go trough procedures and also from Nagasaki Airport to the city. Hyperdia gives about 5 hours of travel time all together (excluding airport procedures), realistically speaking you should add at least 1 more hour.
Hope this helps,
Daniel